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Power for Equivalence in Negative Binomial Rate Ratio

Source: R/RcppExports.R
nbpowerequiv.Rd

Obtains the power for equivalence in negative binomial rate ratio.

Usage

nbpowerequiv(
  kMax = 1L,
  informationRates = NA_real_,
  criticalValues = NA_real_,
  alpha = 0.05,
  typeAlphaSpending = "sfOF",
  parameterAlphaSpending = NA_real_,
  userAlphaSpending = NA_real_,
  rateRatioLower = NA_real_,
  rateRatioUpper = NA_real_,
  allocationRatioPlanned = 1,
  accrualTime = 0L,
  accrualIntensity = NA_real_,
  piecewiseSurvivalTime = 0L,
  stratumFraction = 1L,
  kappa1 = NA_real_,
  kappa2 = NA_real_,
  lambda1 = NA_real_,
  lambda2 = NA_real_,
  gamma1 = 0L,
  gamma2 = 0L,
  accrualDuration = NA_real_,
  followupTime = NA_real_,
  fixedFollowup = FALSE,
  spendingTime = NA_real_,
  studyDuration = NA_real_
)

Arguments

kMax

The maximum number of stages.

informationRates

The information rates. Defaults to (1:kMax) / kMax if left unspecified.

criticalValues

Upper boundaries on the z-test statistic scale for stopping for efficacy.

alpha

The significance level for each of the two one-sided tests. Defaults to 0.05.

typeAlphaSpending

The type of alpha spending. One of the following: "OF" for O'Brien-Fleming boundaries, "P" for Pocock boundaries, "WT" for Wang & Tsiatis boundaries, "sfOF" for O'Brien-Fleming type spending function, "sfP" for Pocock type spending function, "sfKD" for Kim & DeMets spending function, "sfHSD" for Hwang, Shi & DeCani spending function, "user" for user defined spending, and "none" for no early efficacy stopping. Defaults to "sfOF".

parameterAlphaSpending

The parameter value for the alpha spending. Corresponds to \(\Delta\) for "WT", \(\rho\) for "sfKD", and \(\gamma\) for "sfHSD".

userAlphaSpending

The user defined alpha spending. Cumulative alpha spent up to each stage.

rateRatioLower

The lower equivalence limit of rate ratio.

rateRatioUpper

The upper equivalence limit of rate ratio.

allocationRatioPlanned

Allocation ratio for the active treatment versus control. Defaults to 1 for equal randomization.

accrualTime

A vector that specifies the starting time of piecewise Poisson enrollment time intervals. Must start with 0, e.g., c(0, 3) breaks the time axis into 2 accrual intervals: \([0, 3)\) and \([3, \infty)\).

accrualIntensity

A vector of accrual intensities. One for each accrual time interval.

piecewiseSurvivalTime

A vector that specifies the starting time of piecewise exponential survival time intervals. Must start with 0, e.g., c(0, 6) breaks the time axis into 2 event intervals: \([0, 6)\) and \([6, \infty)\). Defaults to 0 for exponential distribution.

stratumFraction

A vector of stratum fractions that sum to 1. Defaults to 1 for no stratification.

kappa1

The dispersion parameter (reciprocal of the shape parameter of the gamma mixing distribution) for the active treatment group by stratum.

kappa2

The dispersion parameter (reciprocal of the shape parameter of the gamma mixing distribution) for the control group by stratum.

lambda1

The rate parameter of the negative binomial distribution for the active treatment group by stratum.

lambda2

The rate parameter of the negative binomial distribution for the control group by stratum.

gamma1

The hazard rate for exponential dropout, a vector of hazard rates for piecewise exponential dropout applicable for all strata, or a vector of hazard rates for dropout in each analysis time interval by stratum for the active treatment group.

gamma2

The hazard rate for exponential dropout, a vector of hazard rates for piecewise exponential dropout applicable for all strata, or a vector of hazard rates for dropout in each analysis time interval by stratum for the control group.

accrualDuration

Duration of the enrollment period.

followupTime

Follow-up time for the last enrolled subject.

fixedFollowup

Whether a fixed follow-up design is used. Defaults to FALSE for variable follow-up.

spendingTime

A vector of length kMax for the error spending time at each analysis. Defaults to missing, in which case, it is the same as informationRates.

studyDuration

Study duration for fixed follow-up design. Defaults to missing, which is to be replaced with the sum of accrualDuration and followupTime. If provided, the value is allowed to be less than the sum of accrualDuration and followupTime.

Value

An S3 class nbpowerequiv object with 4 components:

  • overallResults: A data frame containing the following variables:

    • overallReject: The overall rejection probability.

    • alpha: The overall significance level.

    • numberOfEvents: The total number of events.

    • numbeOfSubjects: The total number of subjects.

    • exposure: The total exposure.

    • studyDuration: The total study duration.

    • information: The maximum information.

    • expectedNumberOfEvents: The expected number of events.

    • expectedNumberOfSubjects: The expected number of subjects.

    • expectedExposure: The expected exposure.

    • expectedStudyDuration: The expected study duration.

    • expectedInformation: The expected information.

    • kMax: The number of stages.

    • rateRatioLower: The lower equivalence limit of rate ratio.

    • rateRatioUpper: The upper equivalence limit of rate ratio.

    • rateRatio: The rate ratio.

    • accrualDuration: The accrual duration.

    • followupTime: The follow-up duration.

    • fixedFollowup: Whether a fixed follow-up design is used.

  • byStageResults: A data frame containing the following variables:

    • informationRates: The information rates.

    • efficacyBounds: The efficacy boundaries on the Z-scale for each of the two one-sided tests.

    • rejectPerStage: The probability for efficacy stopping.

    • cumulativeRejection: The cumulative probability for efficacy stopping.

    • cumulativeAlphaSpent: The cumulative alpha for each of the two one-sided tests.

    • cumulativeAttainedAlphaH10: The cumulative alpha attained under H10.

    • cumulativeAttainedAlphaH20: The cumulative alpha attained under H20.

    • numberOfEvents: The number of events.

    • numberOfDropouts: The number of dropouts.

    • numberOfSubjects: The number of subjects.

    • exposure: The exposure.

    • analysisTime: The average time since trial start.

    • efficacyRateRatioLower: The efficacy boundaries on the rate ratio scale for the one-sided null hypothesis at the lower equivalence limit.

    • efficacyRateRatioUpper: The efficacy boundaries on the rate ratio scale for the one-sided null hypothesis at the upper equivalence limit.

    • efficacyP: The efficacy bounds on the p-value scale for each of the two one-sided tests.

    • information: The cumulative information.

  • settings: A list containing the following input parameters: typeAlphaSpending, parameterAlphaSpending, userAlphaSpending, allocationRatioPlanned, accrualTime, accuralIntensity, piecewiseSurvivalTime, stratumFraction, kappa1, kappa2, lambda1, lambda2, gamma1, gamma2, spendingTime.

  • byTreatmentCounts: A list containing the following counts by treatment group:

    • numberOfEvents1: The number of events by stage for the treatment group.

    • numberOfDropouts1: The number of dropouts by stage for the treatment group.

    • numberOfSubjects1: The number of subjects by stage for the treatment group.

    • exposure1: The exposure by stage for the treatment group.

    • numberOfEvents2: The number of events by stage for the control group.

    • numberOfDropouts2: The number of dropouts by stage for the control group.

    • numberOfSubjects2: The number of subjects by stage for the control group.

    • exposure2: The exposure by stage for the control group.

    • expectedNumberOfEvents1: The expected number of events for the treatment group.

    • expectedNumberOfDropouts1: The expected number of dropouts for the treatment group.

    • expectedNumberOfSubjects1: The expected number of subjects for the treatment group.

    • expectedExposure1: The expected exposure for the treatment group.

    • expectedNumberOfEvents2: The expected number of events for control group.

    • expectedNumberOfDropouts2: The expected number of dropouts for the control group.

    • expectedNumberOfSubjects2: The expected number of subjects for the control group.

    • expectedExposure2: The expected exposure for the control group.

See also

nbstat

Author

Kaifeng Lu, kaifenglu@gmail.com

Examples


# Example 1: Variable follow-up design
nbpowerequiv(kMax = 2, informationRates = c(0.5, 1),
             alpha = 0.05, typeAlphaSpending = "sfOF",
             rateRatioLower = 2/3, rateRatioUpper = 3/2,
             accrualIntensity = 1956/1.25,
             kappa1 = 5, kappa2 = 5,
             lambda1 = 0.125, lambda2 = 0.125,
             gamma1 = 0, gamma2 = 0,
             accrualDuration = 1.25,
             followupTime = 2.75, fixedFollowup = FALSE)
#>                                                                                       
#> Group-sequential design with 2 stages for equivalence in negative binomial rate ratio 
#> Lower limit for rate ratio: 0.667, upper limit for rate ratio: 1.5, rate ratio: 1     
#> Event rate for treatment: 0.125, event rate for control: 0.125                        
#> Dispersion for treatment: 5, dispersion for control: 5                                
#> Overall power: 0.8995, overall alpha: 0.05                                            
#> Maximum # events: 825.2, expected # events: 825.2                                     
#> Maximum # subjects: 1956, expected # subjects: 1956                                   
#> Maximum exposure: 6601.5, expected exposure: 6601.5                                   
#> Maximum information: 66.18, expected information: 66.18                               
#> Total study duration: 4, expected study duration: 4                                   
#> Accrual duration: 1.2, follow-up duration: 2.8, fixed follow-up: FALSE                
#> Allocation ratio: 1                                                                   
#> Alpha spending: Lan-DeMets O'Brien-Fleming                                            
#>                                                                                       
#>                                        Stage 1 Stage 2
#> Information rate                       0.500   1.000  
#> Boundary for each 1-sided test (Z)     2.538   1.662  
#> Cumulative rejection                   -0.0000 0.8995 
#> Cumulative alpha for each 1-sided test 0.0056  0.0500 
#> Cumulative alpha attained under H10    0.0000  0.0500 
#> Cumulative alpha attained under H20    0.0000  0.0500 
#> Number of events                       213.1   825.2  
#> Number of dropouts                     0.0     0.0    
#> Number of subjects                     1956.0  1956.0 
#> Exposure                               1705.2  6601.5 
#> Analysis time                          1.5     4.0    
#> Boundary for lower limit (rate ratio)  1.036   0.818  
#> Boundary for upper limit (rate ratio)  0.965   1.223  
#> Boundary for each 1-sided test (p)     0.0056  0.0482 
#> Information                            33.09   66.18  

# Example 2: Fixed follow-up design
nbpowerequiv(kMax = 2, informationRates = c(0.5, 1),
             alpha = 0.05, typeAlphaSpending = "sfOF",
             rateRatioLower = 0.5, rateRatioUpper = 2,
             accrualIntensity = 220/1.5,
             stratumFraction = c(0.2, 0.8),
             kappa1 = 3, kappa2 = 3,
             lambda1 = c(8.4, 10.2),
             lambda2 = c(8.0, 11.5),
             gamma1 = 0, gamma2 = 0,
             accrualDuration = 1.5,
             followupTime = 0.5, fixedFollowup = TRUE)
#>                                                                                       
#> Group-sequential design with 2 stages for equivalence in negative binomial rate ratio 
#> Lower limit for rate ratio: 0.5, upper limit for rate ratio: 2, rate ratio: 0.917     
#> Stratum fraction: 0.2 0.8                                                             
#> Event rate for treatment: 8.4 10.2, event rate for control: 8 11.5                    
#> Dispersion for treatment: 3, dispersion for control: 3                                
#> Overall power: 0.7485, overall alpha: 0.05                                            
#> Maximum # events: 1135.2, expected # events: 1135.2                                   
#> Maximum # subjects: 220, expected # subjects: 220                                     
#> Maximum exposure: 110, expected exposure: 110                                         
#> Maximum information: 17.2, expected information: 17.2                                 
#> Total study duration: 2, expected study duration: 2                                   
#> Accrual duration: 1.5, follow-up duration: 0.5, fixed follow-up: TRUE                 
#> Allocation ratio: 1                                                                   
#> Alpha spending: Lan-DeMets O'Brien-Fleming                                            
#>                                                                                       
#>                                        Stage 1 Stage 2
#> Information rate                       0.500   1.000  
#> Boundary for each 1-sided test (Z)     2.538   1.662  
#> Cumulative rejection                   -0.0000 0.7485 
#> Cumulative alpha for each 1-sided test 0.0056  0.0500 
#> Cumulative alpha attained under H10    0.0000  0.0500 
#> Cumulative alpha attained under H20    0.0000  0.0500 
#> Number of events                       427.2   1135.2 
#> Number of dropouts                     0.0     0.0    
#> Number of subjects                     119.5   220.0  
#> Exposure                               41.4    110.0  
#> Analysis time                          0.8     2.0    
#> Boundary for lower limit (rate ratio)  1.188   0.746  
#> Boundary for upper limit (rate ratio)  0.842   1.340  
#> Boundary for each 1-sided test (p)     0.0056  0.0482 
#> Information                            8.60    17.20